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The prompt
The prompt
Mathematical inquiry processes: Test cases; conjecture, generalise and prove. Conceptual field of inquiry: Transformation matrices; matrix multiplication.
The matrix multiplication prompt was designed for an A-level Further Mathematics class. The 12 students had already studied matrix arithmetic and how to find an inverse. In the orientation phase of the inquiry, the students posed questions about the prompt and discussed the conditions under which it might be true:
Are there two transformations or more than two?
Are there matrices for all the transformations - reflection, rotation, translation and enlargement?
Do you add or multiply two matrices to represent two transformations?
If you rotate and then enlarge a shape, then the result is different to enlarging and rotating the shape.
If you call the transformations A and B, then AB = BA in the case of transformations - that is, order does not matter.
Order does matter because AB is not equal to BA.
Whether AB equals BA or not depends on whether the shape is 'regular' (with one or more line of symmetry).
We should test different cases by using shapes with and without lines of symmetry.
It depends on the order of the transformations because you might not be able to multiply the matrices.
Preliminary inquiry
To answer the question about whether there are matrices that represent all the transformations, the teacher might choose to carry out a preliminary inquiry (see the PowerPoint in the Resources below). Students determine what each matrix represents.
Note: Translations can be represented by matrix addition and subtraction. The teacher is advised to exclude translations from the main inquiry after an initial discussion or explanation of how the matrices work.
Testing cases
After the orientation phase, the class chose to test cases in order to verify a rule. Is the single matrix that represents a combined transformation the product of AB or BA or could it be both? A few students spontaneously started to develop conjectures, propose generalisations and construct algebraic proofs. We present examples of their inquiries below.
True or false?
If we consider combinations of reflections, rotations (anti-clockwise about the origin), enlargements and shears (both with centres at the origin), then the statement in the prompt is always true. The single matrix in the case where matrix A represents the first transformation and matrix B the second is BA.
January 2023
Classroom inquiry
Classroom inquiry
1. Test different cases
2. Conjecture and generalise
3. Prove
Conjecture and prove
Conjecture and prove
As his Further Maths class discussed the prompt, Qiang, a year 12 student, was thinking on his own. When it was his turn to feed back, he hesitantly put forward a conjecture: All transformation matrices are square in n dimensions. While his peers were exploring, Qiang set about proving his conjecture. He wrote up his notes (see below) for the next lesson and explained them to the class.
In the final section of the notes, Qiang has drawn on the concept of extrusion from computer graphics. He explained that extrusion means to render a 2-dimensional object in three dimensions in a similar manner to turning a square into a cube.
February 2024
Resources
Resources
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